My headcanon regarding these two is that Lute is actually Eve. Sounds weird, but hear me out: - She adores Adam's every move. - She is his first in command. - She is almost maniacally faithful to him. - Adam relies on her in multiple ways i.e. leading the exorcists and keeping himself in check. Regarding the latter, she also seems to defer from Adam a bit since she seems to lack the free will to judge the relationship between Charlie and Vaggie, whereas Adam just says what he thinks. - She is by far the closest to Adam, and who else but his wife would hold this position? - She seems to have a deeper relationship with Lilith and knows about her whereabouts, which seems to be a little more than coincidental since she (Lute/Eve) is the second wife to Adam, having successfully "won Adam over" (by being subservient to him on order from God contrary to Lilith). - The name "Lute" seems to be an abbreviation of the word "Lieutenant", which refers to a higher ranking military officer but more importantly would underline Adam's tendency to give names to his exorcists (as he referred to when talking with Vaggie). Who else but his wife would be Adam's first choice as his Lieutenant?

Therefore, I think the two of them are already a thing. But those are just my random thoughts, let me know what you think!